Ultrastrong Coupling of a Single Molecule to a Plasmonic Nanocavity: A First-Principles Study

Ultrastrong coupling (USC) is a distinct regime of light-matter interaction in which the coupling strength is comparable to the resonance energy of the cavity or emitter. In the USC regime, common approximations to quantum optical Hamiltonians, such as the rotating wave approximation, break down as the ground state of the coupled system gains photonic character due to admixing of vacuum states with higher excited states, leading to ground-state energy changes. USC is usually achieved by collective coherent coupling of many quantum emitters to a single mode cavity, whereas USC with a single molecule remains challenging. Here, we show by time-dependent density functional theory (TDDFT) calculations that a single organic molecule can reach USC with a plasmonic dimer, consisting of a few hundred atoms. In this context, we discuss the capacity of TDDFT to represent strong coupling and its connection to the quantum optical Hamiltonian. We find that USC leads to appreciable ground-state energy modifications accounting for a non-negligible part of the total interaction energy, comparable to kBT at room temperature.


Figures
: Photoabsorption spectra. Additional photoabsorption spectra of metal dimers strongly coupled to tuned molecules (solid lines), gray dotted lines mark reference spectra of the metal dimers and molecules alone: (a) Mg 201 nanodimers coupled to two tetracene molecules, (b) Mg 309 dimers coupled to 1, 3, and 6 tetracene molecules, (c) Al 309 dimers with 1, 2, 3, and 6 benzene molecules.     Tables   Table S1: Summary of modelled systems. List of modelled systems sorted by atom, structure, gap size, molecule and number N . The molecules are placed in the center of the gap between the dimer components. For Mg and Na for N = 2 the molecules are placed parallel to each other in the center of the dimer gap and are separated by 2Å to ensure a slightly red detuned transition with respect to the plasmon. For Al for N > 2 the molecules are placed between the corners of the parallel facets and for N = 6 also between the edges.

Supplementary Notes
Supplementary Note S1: Mode profiles and induced fields of Mg 201 dimers. The amplifying effect of the molecule on the coupling strength efficiency is supported by the induced fields presented in Fig. S4. Specifically, the fact that the molecule couples to the cavity mode with greater efficiency than predicted by the vacuum field of the bare cavity is confirmed by an analysis of the mode profiles and cross sections of the enhanced induced electric field plotted in Fig. S4 for Mg 201 dimers with 15 and 30Å gaps. In the former case in Fig. S4a, the mode is confined to the gap, but its amplitude is on the order of 0.3-0.4 and only very close to the dimers does it exceed 0.5. However, for two naphthalene molecules we obtain effective vacuum fields which are ca. 3 times larger than the theoretical one and ca. 2 times larger for for tetracene (ratio of the effective and theoretical mode volumes plotted in Fig. 4). In all cases the effective vacuum fields are higher than the ones based on the QNMs.
These increases of the effective vacuum fields are the result of a modification of the cavity's mode volume due to the molecule [1]. This change can be understood in terms of a logical split of the molecule into the electronic HOMO-LUMO transition, which couples to the gap plasmon, and the higher-energy transitions which modify the cavity mode. The result of this can be illustrated by plotting the induced electric field enhancements in Fig. S4 for the bare dimers at the cavity resonance and at the upper and lower polaritons. The addition of molecules to the gap modifies the electric field distribution by focusing its intensity around the molecule. This amplifies the energy density in the gap and results in enhanced coupling beyond what is predicted based on the mode profile of the empty cavity [1]. For the small gap of 15Å the field enhancement is larger for naphthalene in the gap than for tetracene, what is consistent with the larger amplification of the vacuum vield for the former case. In the 30Å-gap dimer the field enhancement for both molecules is similar, what is consistent with the equal effective vacuum fields. These results highlight the importance of considering the molecules themselves as equivalent participants to nano-and picoscale cavities in coupling phenomena. Indeed, we observe that molecules may alter small cavities to such large degree, that their size may begin to contribute a large or dominant part of a cavity's mode volume, Fig. S7.

Supplementary Note S2: Longitudinal QED Hamiltonian.
Electrostatic interaction. We consider the system of a single molecule placed at the center of a dimer nanocluster gap (see Fig. S9). The general Hamiltonian description of this system is the minimal coupling Hamiltonian [2]: where p i , m i , r i are the momenta, masses and positions of single charges and V C is the Coulomb potential taking longitudinal, short-range electromagnetic interaction between charges into account. The effect of transverse electromagnetic radiation is here accounted by the potential vector A ⊥ (r i ), which induces a coupling term and a self-energy term with the transverse field. However, as risen in refs. [3,4], the effects of transverse fields should arise when the size of the system is comparable toλ = c/E, E being the resonant energy. For 1 < E < 10 eV, we have 124 <λ < 1240 nm, with in our case is always much larger than the size of the dimer. We can then neglect the effects of the transverse radiation and the Hamiltonian reduced to: When considering our two-moieties system (molecule and NP dimer) with only induced dipole moments, one can decompose the Hamiltonian into: where ck labels the kth nanocluster, m labels the molecule, and the Coulomb interaction V C is split into "self" terms summed over charges within each moiety and an interacting term V int including interactions between charges of different moieties. Figure S9: System of a single molecule (as an example: benzene) located in the gap center of a nanoparticle dimer. The gap of length d is defined as the minimal surface-to-surface distance in between the nanoparticles. Below: effective 3-point-dipole modeling of the system. The molecule's point-dipole with moment µ m is placed between two identical point-dipoles associated with each nanoparticle, with moments µ c . The effective distance d is larger than d but smaller than d + 2R, R being the radius of a single nanoparticle.
If we now assume a dipolar polarization for the nanoclusters and the molecule, we can map the problem to a three-coupled-dipole system where each cluster has an effective dipole moment µ c (see fig. S9). Modeling the charge oscillations as harmonic oscillators, the Hamiltonian becomes: where here ω c is the resonance frequency of a single nanocluster with operators obeying [p k , p † k ] = δ kk , ω m the transition frequency of the molecule with operators obeying [b, b † ] = 1, and we have decomposed the induced dipole Coulomb interaction terms V αβ , α, β = c1, c2, m: where R αβ = r α − r β , r α being the position of the dipole α. Considering now the simple situation where all dipoles are aligned, the molecule is placed at the origin and each cluster dipole is placed at a distance d /2 of the molecule, the Hamiltonian reads: This expression enables to isolate the effective dimer Hamiltonian and diagonalize it to derive the effective coupling strength between the dimer and the molecule.
Diagonalization of the dimer part. Considering now only the dimer part: we can define the Hopfield operators [5]: with the Hopfield coefficients satisfying the condition |x ± | 2 − |y ± | 2 + |m ± | 2 − |h ± | 2 = 1 to ensure the bosonicity of the new operators. The latter enters the eigenvalue problem [Π ± , H d ] = ω ± Π ± that can be stated in a matrix form: where we set g c = −µ 2 c /(π 0 d 3 ). Diagonalizing the Hopfield matrix analytically yields the Hopfield coefficients as well as the eigenfrequencies: (S10) These two eigenfrequencies corresponds to two eigenmodes: the lower energy mode ω − is the one described in fig. S9, where the two dipoles are aligned and oscillate in phase (| + − + − ) while the higher energy mode is a "dark" contribution where the dipoles oscillates with a phase of π (| − + + − ). Therefore, we identify the dimer eigenfrequency as ω d ≡ ω − , which is the peak seen in e.g. photoabsorption spectra. It is then expected that ω d < ω c , which is clearly seen by looking at e.g. an Al 309 cluster (ω c ≈ 7.7 eV [6]) and a 2Al 309 dimer (ω d ≈ 7.2 eV). Using (S10), we can even deduce the coupling strength g c between the clusters as: In addition, if the single cluster dipole moment µ c is known, one can then find the effective distance d between the two point dipoles associated with each cluster. The remaining part of the diagonalization procedure consists in inverting the transformation: where a ± = x ± + y ± and b ± = m ± + h ± . Solving for the Hopfield coefficients, we find a ± = ω ± /(2ω c ) and b ± = ± ω ± /(2ω c ). This allows for writing the operator p † 1 + p 1 − p † 2 − p 2 in terms of the new operators in (S6) and we find: Eigenfrequencies of the dimer-molecule system. Considering the case where the molecule is nearly resonant with the bright dimer mode ω d ≡ ω − , the dark mode can be neglected and we have: where B ≡ Π − and g md = − µcµm 2ωc ω d is the coupling strength between the bright mode and the molecule exciton. This Hamiltonian can again be diagonalized by introducing new polaritonic operators: with the new Hopfield coefficients satisfying |ξ ± | 2 − |υ ± | 2 + |µ ± | 2 − |ν ± | 2 = 1. The diagonal form is: with the analytical eigenfrequencies being given by the biquadratic equation: yielding: The ground state modification is then given by: (S19) When the system is on resonance ω m = ω d ≡ ω, then: The first correction to the ground state energy shift is then: which then corresponds to the London force (van der Waals between induced dipoles) in 1/R 6 [4]. The van der Waals interaction can then be interpreted from the point of view of USC regime physics.
Supplementary Note S3: Canonical Transformation of Subsystems.
The direct random phase approximation Hamiltonian (see e.g. ref. [7]) coupling plasmonic (P ) and molecular (M ) subsystems iŝ where ∆ ia,i a = δ ii δ aa ( a − i ) contains non-interacting Kohn-Sham excitation energies in its diagonal and K ia,i a = dr dr ψ i (r)ψ a (r)ψ i (r )ψ a (r ) 4π 0 |r − r | (S23) contains the Coulomb matrix elements. We can first transform into generalized position and momentum representation, We then diagonalize the Casida equations for the individual subsystems as O T P Ω 2 P O P = ∆ 2 P + 2 √ ∆ P K P P √ ∆ P , for M respectively and perform a following canonical transformation which results into a following Hamiltonian matrix closer by adding indices. Let I P and I M enumerate Casida eigenvectors for plasmon and molecular system respectively, and k P and k M indices stand for the Kohn-Sham state pairs of plasmon and molecular system respectively, and assuming dipolar coupling via the dipole tensor V cc (R), where R is the distance vector between the subsystems and c ∈ {x, y, z}, The transition dipole moment of a Casida transition is and thus we may write Eq. S30 as (K P M ) I P I M = cc µ I P ,c V cc (R)µ I M ,c , and so we have finally established that subsystems may first be diagonalized, and subsequently coupled, while the only approximation performed is the dipole approximation. Our recent work [8] on dipole coupling plasmonic nanoparticles utilizes this by mutually coupling already diagonalized TDDFT systems. In other words, Casida excitation energies and transition dipole moments are sufficient to exactly couple two subsystems (up to the dipole approximation). Renormalization inside the subsystem sub-block for example due to plasmon lifting the excitation energies are fully accounted for by the diagonalized Casida eigenvectors and dipole moments. There are no special intersystem Coulomb effects beyond the information encoded into the Casida eigenvectors.